The figure starts with a golden rectangle ABCD,
with AB = 1 and BC = j = (1 +
Ö5)/2 = the
golden ratio; we draw the square ABA'D' and get the golden rectangle
A'B'C'D' the length of the sides of which are 1/j
and 1.This process is repeated to get the rectangle A''B''C''D''
and so on and so forth, as well as the external squares; the golden
spiral is composed of consecutive quarters of circles inscribed in
each square.

The dotted diagonals BD and CD' intersect
at the point O, which is the asymptotic point of the spiral, called
"eye of God" (the golden ratio being also called the "divine proportion"!)

With each turn, the
radius of the golden spiral is multiplied by ;
the constant polar tangential angle is equal to

The golden spiral (in red) and the true logarithmic spiral (in green)

If, instead of starting from a golden rectangle, we begin
with a Fibonacci rectangle, with AB = Fn-1
and BC = Fn, where Fn-1
and Fn are two consecutive Fibonacci
numbers, we get a spiral composed of quarter of circles called Fibonacci
spiral which approximates the golden spiral, but does not have the
"eadem mutata resurgo" property.

We can also begin with a golden triangle (isosceles triangle
with angle at the vertex equal to p/5). We get
a spiral that can be called triangular golden spiral; when this
spiral is turned by 3p/5, it is enlarged by
a factor j ; therefore, it is an approximation
of the logarithmic spiral ;
the enlargement factor at each turn is
and the polar tangential angle is about 76°.

See also the Padovan
spiral, the enlargement factor at each turn of which is the sixth power
of the Padovan number, i.e. about 5.4.

On this
site, can be found an explanation of the fact that the shell of the
nautilus supposedly approximates a golden spiral, but this is obviously
contradicted by a direct observation: the spiral of the nautilus is much
tighter and experimental measurements show that its enlargement factor
is about 3 which is clearly less than the 6.9 of the golden spiral...

Ammonite: clearly smaller enlargement factor....

The tendril of the green bean supposedly approximates
the triangular golden spiral...

Compare the construction of the golden spiral to that
of the spiral with n
centres, which approximates an Archimedean spiral.